Kudos on the tests you've used to eliminate some of the possibilities.
Since you've eliminated them, my money is on the radio.
This is just a wild guess, but I'll bet there's a preamp in the radio circuitry that's not used in the CD player circuit. These units are no longer made to be repaired, and even if it could be I doubt if you could find a repair shop to do it. If you did, it would probably cost a lot more to repair it than to replace the radio. You can buy a new unit for the cost of one (perhaps two) hours of shop time. Individual parts probably couldn't even be purchased by a tech. An individual part used to be a wirewound resister, or a capacitor, or a transistor, or something of that sort. Now it's all in thin-film "chips". And the frame structures are now all ultrasonically welded plastics or "pop in" parts with little plastic bifurcated "mushrooms" that cannot be disassembled nondestructively once popped together.
NOTE" should you decide to replace the unit yourself, I strongly urge you to spend the extra $20 on an "adapter cable". This is spliced onto the new unit's leads on the kitchen table following easy-to-follow directions and then plugs directly into your OEM wiring harness. Installation of the radio is much, much easier and the problems often associated with butchered wiring harnesses are avoided. If you have a shop install the new unit, INSIST that they use an adapter cable. If they give you and argument, go elsewhere.
I've installed a few radios and CD players over the years, and there's absolutely no way I'd do it (or allow anyone else to) without an adapter cable.